Abstract
We prove Jackson, realization, and converse theorems for Freud weights inL p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL p for 0<p ≤ ∞. Since theK-functional is identically zero inL p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.
Similar content being viewed by others
References
B. Bojanov (to appear):A Jackson type theorem for Chebyshev systems. Math. Balkanica.
J. Clunie, T. Kövari (1968):On integral functions having prescribed asymptotic growth, II. Canad. J. Math.,20:7–20.
Z. Ditzian, V. Totik (1987): Moduli of Smoothness. Springer Series in Computational Mathematics, Vol. 9. Berlin: Springer-Verlag.
Z. Ditzian, V. H. Hristov, K. G. Ivanov (1995):Moduli of smoothness and K-functionals in L p, 0<p<1. Constr. Approx.,11:67–83.
R. A. Devore, V. A. Popov (1988):Interpolation of Besov spaces. Trans. Amer. Math. Soc.,305:397–414.
R. A. Devore, D. Leviatan, X. M. Yu (1992):Polynomial approximation in L p (0<p<1). Constr. Approx.,8:187–201.
G. Freud (1977):On Markov-Bernstein type inequalities and their applications. J. Approx. Theory,19:22–37.
G. Freud, H. N. Mhaskar (1983):K-Functionals and moduli of continuity in weighted polynomial approximation. Arkiv. Matematik,21:145–161.
V. H. Hristov, K. G. Ivanov (1990):Realization of K-functionals on subsets and constrained approximation. Math. Balkanica, 4 (New Series),3:236–257.
K. G. Ivanov, V. Totik (1990):Fast decreasing polynomials. Constr. Approx.,6:1–20.
P. Koosis (1988): The Logarithmic Integral I. Cambridge Advanced Studies in Mathematics, Vol. 12. Cambridge: Cambridge University Press.
P. Koosis (1992): The Logarithmic Integral II. Cambridge Advanced Studies in Mathematics, Vol. 21. Cambridge: Cambridge University Press.
A. Kroo, J. Szabados (1995):Weighted polynomial approximation on the real line. J. Approx. Theory,83:41–64.
D. Leviatan, X. M. Yu (1993): Shape Preserving Approximation by Polynomials inL p. Manuscript.
A. L. Levin, D. S. Lubinsky (1990):L ∞ Markov and Bernstein inequalities for Freud weights. SIAM J. Math. Anal.,21:1065–1082.
A. L. Levin, D. S. Lubinsky (1992):Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for Freud weights. Constr. Approx.,8:463–535.
A. L. Levin, D. S. Lubinsky (1994):L p Markov-Bernstein inequalities for Freud weights. J. Approx. Theory,77:229–248.
G. G. Lorentz (1986): Approximation of Functions, 2nd edn. New York: Chelsea.
D. S. Lubinsky (1989): Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdös Type Weights. Pitman Research Notes in Mathematics, Vol. 202. Harlow, Essex: Longman.
D. S. Lubinsky, P. Nevai (1987):Markov-Bernstein inequalities revisited. Chinese J. Approx. Theory Appl.,3(4):98–119.
D. S. Lubinsky, E. B. Saff (1988): Strong Asymptotics for Extremal Polynomials Associated with Exponential Weights. Lecture Notes in Mathematics, Vol. 1305. Berlin: Springer-Verlag.
H. N. Mhaskar, E. B. Saff (1984):Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc.,285:203–234.
H. N. Mhaskar, E. B. Saff (1985):Where does the sup-norm of a weighted polynomial live? Constr. Approx.,1:71–91.
H. N. Mhaskar, E. B. Saff (1987):Where does the L p norm of a weighted polynomial live? Trans. Amer. Math. Soc.,303:109–124.
P. Nevai (1979): Orthogonal Polynomials. Memoirs of the American Mathematical Society, No. 213. Providence, RI: American Mathematical Society.
P. Nevai (1986):Geza-Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory,48:3–167.
P. Nevai, ed. (1990): Orthogonal Polynomials, Theory and Practice. NATO ASI Series, Vol. 294. Dordrecht: Kluwer.
P. Nevai, P. Vertesi (1985):Mean convergence of Hermite-Fejer interpolation. J. Math. Anal. Appl.,105:26–58.
P. P. Petrushev, P. Popov (1987): Rational Approximation of Real Functions. Cambridge: Cambridge University Press.
E. B. Saff, V. Totik (to appear): Logarithmic Potential with External Fields. New York: Springer-Verlag.
U. Schmid (1992):On power series with non-negative coefficients. Complex Variables,18:187–192.
Author information
Authors and Affiliations
Additional information
Communicated by Dietrich Braess.
Rights and permissions
About this article
Cite this article
Ditzian, Z., Lubinsky, D.S. Jackson and smoothness theorems for freud weights. Constr. Approx 13, 99–152 (1997). https://doi.org/10.1007/BF02678431
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02678431